If a Moon Druid is wildshaped into a Giant Spider, which has a web attack (which is classified by the MM as a ranged weapon attack), can he then use the Sharpshooter feat to gain a +10 to damage since it's a ranged weapon attack anyway?

I'll boil it down for you. When we wrote the "Player's Handbook," we meant the weapons on the weapon table when we wrote "weapon," unless we said otherwise. But you break nothing in the game if you let natural weapons go along for the ride.